Aircraft: CESSNA 337 SUPER SKYMASTER
Phase of flight: Cruise
Anomaly: Airspace Violation All Types; Deviation - Altitude Overshoot; Deviation / Discrepancy - Procedural FAR; Deviation / Discrepancy - Procedural Clearance; Inflight Event / Encounter Weather / Turbulence; Inflight Event / Encounter VFR In IMC

Was VFR enroute to BZN at 15500 feet with flight following talking to SLC Center on 134.1. I contacted SLC Center and communicated I was going to climb to 17500 feet as I was approaching clouds ahead. As I was climbing up over the clouds my climb was accelerating upward due to convective activity and I was contacted by SLC who saw that I had climbed above 17500 and in fact had exceeded 18000 and was therefore in Class A airspace. I saw that it looked like I could descend back lower heading to the west but I decided I should request an IFR clearance to be able to fly in Class A airspace and climb higher. Before I could call I was contacted by SLC and told that I was to contact the military frequency for Jarbidge and was given a clearance direct to Mountain Home. I turned to the left toward Mountain home and contacted the military facility. I believe the frequency was 118.05. I contacted the military facility and was told that I needed to descend to a lower altitude. I told the controller that it looked like I would be able to descend up ahead. I failed to tell the controller about the convective activity I was experiencing and that I was above the clouds. However due to the aggressive convective activity of the storm building beneath me; I then asked the controller for an IFR clearance to BZN and was told that this was not possible within the military airspace that I was now in. I was called back by the military controller and told to contact SLC Center again and they were going to work something out for me. I contacted SLC Center again and I was given an IFR clearance to 21000 to BZN with vectors. I was given a heading of 070 which I could see was to fly me to the east out of the MOA. However I replied that I was unable to turn to the east due to a large buildup/ thunderstorm to the east on my right side. I continued to turn east as much I was able to and soon got on a heading of 070. In the process; the convective activity pushed me up beyond 21000 and I was contacted to get back down to my assigned altitude. In the middle of dealing with maneuvering around the thunderstorm to the east during the climb and the convective activity my engines were overheating due to the decreased cooling effect of the thinner air. At the same time all of this was going on I was managing changes to cowl flaps; mixture; power settings and airspeed to get the engine CHT temperatures under control. In the middle of this the convective activity was now pushing me down lower and I requested to descend to 19000. The convective activity carried me down below 19000 and I was again contacted and had to climb back up to the assigned altitude of 19000. I could see that I was staying south of restricted area R3204A and B as I exited the MOA.I realized that I should have executed a 180 degree turn when I was pushed above 17500 feet to get out of the convective area that I had flown into. Then I could have requested an IFR clearance up to a higher altitude prior to beginning my flight. I had in fact filed an IFR flight along the route I was flying and should have activated that flight plan when I took off so that I could have easily requested a change in altitude when I encountered the bad weather south of TWF. I am very thankful for the assistance of the controllers that helped me manage this difficult weather situation that I flew into. I plan to get more training to learn how to handle higher altitude VFR to IFR transitions. My experience with IFR flight at high altitudes has been new learning to fly the pressurized Skymaster. I also realize now that military controllers are not trained to handle IFR traffic for separation since their job is to manage military aircraft that are trying to intercept one another. Cessna Skymaster pilot reported inadvertently climbing VFR into Class A airspace due to convective activity. He negotiated with the Center Controller and a Military Controller for a new IFR clearance away from the convective activity; restricted airspace; and out of the MOA to his destination.